On Fatou’s theorem

Fatou’s theorem (1906) states that a bounded analytic function in the unit disc of the complex plane has radial limits a.e. on the unit circle T . We prove the following.

Theorem 1. Let E be a subset on T. There exists a bounded analytic function in the open unit disc which has no radial limit at each point of E but has unrestricted limit at each point of T\E if and only if E is anFσ set of measure zero.

The necessity of the condition that E is of measure zero follows from Fatou’s theorem. The necessity of the condition that E is an Fσ set is a known elementary (or just obvious) fact. Thus the sufficiency part of Theorem 1 is a converse to Fatou’s theorem. The converses to Fatou’s theorem proved by N. N. Lusin (1919) and by S. V. Kolesnikov (1994) are well-known. The method of the proof of Theorem 1 implies the first elementary and a few line proof for Lusin’s theorem, the original proof of which is very difficult. The same method also simplifies the proof of Kolesnikov’s theorem. Also, the sufficiency part of Theorem 1 immediately implies a well-known theorem of A. J. Lohwater and G. Piranian of 1957.